Question

A coffee shop in Victoria wants to know whether there is a significant difference between the number of lattes sold per day at their Yates St. location and their Johnson St. location.

A sample of 31 days at their Yates St. location reveals an average of 22 lattes sold per day with a standard deviation of 5 lattes. A sample of 31 days at their Johnson St. location reveals an average of 19 lattes sold per day with a standard deviation of 3 days.

What is the value of the sample test statistic if a hypothesis test is conducted at the 2% significance level? Assume that the population standard deviations are equal.

Express your answer rounded to two decimal places.

Answer 1

The provided sample means are shown below:

Also, the provided sample standard deviations are:

and the sample sizes are n_1 = 31 and n_2 = 31

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​

Ha: μ1​ ≠ μ2​

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.02, and the degrees of freedom are df = 60 . In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:

Hence, it is found that the critical value for this two-tailed test is t_c = 2.39 , for α=0.02 and df = 60

(3) Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that |t| = 2.865 > t_c = 2.39 , it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0057 , and since p = 0.0057 < 0.02 , it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μ1​ is different than μ2​, at the 0.02 significance level.